8. Gravity 1.Toward a Law of Gravity 2. Universal Gravitation 3. Orbital Motion 4. Gravitational Energy 5. The Gravitational Field

This TV dish points at a satellite in a fixed position in the sky. How does the satellite manage to stay at that position? period = 24 h

Ptolemaic (Geo-Centric) System epicycle equant deferent swf

8.1. Toward a Law of Gravity 1543: Copernicus – Helio-centric theory. 1593: Tycho Brahe – Planetary obs : Galileo – Jupiter’s moons, sunspots, phases of Venus : Kepler’s Laws 1687: Newton – Universal gravitation. Phases of Venus: Size would be constant in a geocentric system.

Kepler’s Laws Explains retrograde motion Mathematica

8.2. Universal Gravitation Newton’s law of universal gravitation : m 1 & m 2 are 2 point masses. r 12 = position vector from 1 to 2. F 12 = force of 1 on 2. G = Constant of universal gravitation = 6.67  10  11 N m 2 / kg 2. Law also applies to spherical masses. m1m1 m2m2 r 12 F 12

Example 8.1. Acceleration of Gravity Use the law of gravitation to find the acceleration of gravity (a) at Earth’s surface. (b) at the 380-km altitude of the International Space Station. (c) on the surface of Mars.  (a) (b) (c) see App.E

TACTICS 8.1. Understanding “Inverse Square” Given Moon’s orbital period T & distance R from Earth, Newton calculated its orbital speed v and hence acceleration a = v 2 / R. He found a ~ g /  Moon-Earth distance is about 60 times Earth’s radius.

Cavendish Experiment: Weighing the Earth M E can be calculated if g, G, & R E are known. Cavendish: G determined using two 5 cm & two 30 cm diameter lead spheres. Gravity is weakest & long ranged always attractive  dominates at large range. EM is strong & long ranged, can be attractive & repulsive  cancelled out in neutral objects. Weak & strong forces: very short-range.

8.3. Orbital Motion Orbital motion: Motion of object due to gravity from another larger body. E.g. Sun orbits the center of our galaxy with a period of ~200 million yrs. Newton’s “thought experiment” Condition for circular orbit Speed for circular orbit Orbital period Kepler’s 3 rd law g = 0 orbit projectiles

Example 8.2. The Space Station The ISS is in a circular orbit at an altitude of 380 km. What are its orbital speed & period? Orbital speed: Orbital period: Near-Earth orbit T ~ 90 min. Moon orbit T ~ 27 d. Geosynchronous orbit T = 24 h.

Example 8.3. Geosynchronous Orbit What altitude is required for geosynchronous orbits? Altitude = r  R E Earth circumference = Earth not perfect sphere  orbital correction required every few weeks.

Elliptical Orbits Orbits of most known comets, are highly elliptical. Perihelion: closest point to sun. Aphelion: furthest point from sun. Projectile trajectory is parabolic only if curvature of Earth is neglected. ellipse

Open Orbits Closed (circle) Closed (ellipse) Open (hyperbola) Borderline (parabola) Mathematica

8.4. Gravitational Energy How much energy is required to boost a satellite to geosynchronous orbit?  U 12 depends only on radial positions.  U = 0 on this path … so  U 12 is the same as if we start here.

Example 8.4. Steps to the Moon Materials to construct an 11,000-kg lunar observatory are boosted from Earth to geosyn orbit. There they are assembled & launched to the Moon, 385,000 km from Earth. Compare the work done against Earth’s gravity on the 2 legs of the trip. 1 st leg: 2 nd leg:

Zero of Potential Energy  Gravitational potential energy E > 0, open orbit Open Closed E < 0, closed orbit Bounded motionTurning point

Example 8.5. Blast Off ! A rocket launched vertically at 3.1 km/s. How high does it go ? Initial state: Final state: Energy conservation: Altitude = r  R E

Escape Velocity Body with E  0 can escape to   Escape velocity Moon trips have v < v esc. Open Closed

Energy in Circular Orbits Circular orbits:   Higher K or v  Lower E & orbit (r). 0 E U K K

Conceptual Example 8.1. Space Maneuvers Astronauts heading for the International Space Station find themselves in the right circular orbit, but well behind the station. How should they maneuver to catch up? 1.Fire rocket backward to decrease energy & drop to lower, & faster orbit. 2.Fire to circularize orbit. 3.After catching up with the station, fire to boost to up to its level. 4.Fire to circularize orbit. Mathematica

energy 0 UGUG Altitude 0  E = K+U = U / 2 U E = K+U < E ( K < K ) h E = K+U = U / 2 U < U K K > K h < h

GOT IT? 8.3. Spacecrafts A & B are in circular orbits about Earth, with B at higher altitude. Which of the statements are true? (a) B has greater energy. (b) B is moving faster. (c) B takes longer to complete an orbit. (d) B has greater potential energy. (e) a larger proportion of B’s energy is potential energy.     

8.5. The Gravitational Field Two descriptions of gravity: 1.body attracts another body (action-at-a-distance) 2.Body creates gravitational field. Field acts on another body. Near Earth: Large scale: Action-at-a-distance  instantaneous messages Field theory  finite propagation of information Only field theory agrees with relativity. near earth in space Great advantage of the field approach: No need to know how the field is produced.

Moon’s tidal (differential) force field near Earth Moon’s tidal (differential) force field at Earth’s surface Mathematica Application: Tide Two tidal bulges Sun + Moon  tides with varying strength. Tidal forces cause internal heating of Jupiter’s moons. They also contribute to formation of planetary rings.